We consider the problem of initiation of a propagating wave in a one-dimensional excitable fibre. In the Zeldovich-Frank-Kamenetsky equation, a.k.a. Nagumo equation, the key role is played by the "critical nucleus" solution whose stable manifold is the threshold surface separating initial conditions leading to initiation of propagation and to decay. Approximation of this manifold by its tangent linear space yields an analytical criterion of initiation which is in a good agreement with direct numerical simulations.PACS numbers: 87.10.+e, 02.90.+p Threshold phenomena are widespread in bistable dissipative systems. If such a system is spatially extended, then fronts switching from one local state to the other can propagate. Propagating fronts, or trigger waves, play important roles in such diverse physical situations as selfheating in metals and superconductors, phase transitions, combustion and other chemical reaction waves, and biological signalling systems [1-6] to name a few. In biology and chemistry they often appear as a fast stage of pulse waves in "excitable systems" [3,[6][7][8]. The question of existence of such waves in particular mathematical models is well studied. However, whether a propagating wave will actually be observed depends on initial conditions. Understanding conditions of initiation of propagating fronts or pulses is very important in applications. In heart such waves trigger coordinated contraction of the muscle and failure of initiation can cause or contribute to serious or fatal medical conditions, or render inefficient the work of pacemakers or defibrillators [9]. In combustion, understanding of initiation is of critical importance for safety in storage and transport of combustible materials [10].Mathematically, after the external initiating stimulus has finished, the problem is reduced to classification of initial conditions that will or will not lead to a propagating wave solution. This problem is difficult as it is essentially non-stationary, spatially extended and nonlinear, and does not have any helpful symmetries. Yet the problem is so important that analytical answers are highly desirable even if not very accurate.Early attempts of analytical treatment of the initiation problems, including the spatially extended ones, used linear description supplemented with heuristic conditions to represent the threshold [11][12][13][14][15] and, more recently, lowdimensional Galerkin style approximations of the partial differential equations [16,17] In the last two decades, this problem has been analysed from the dynamical systems theory viewpoint [16][17][18][19][20][21][22]. These studies identified the importance of certain "critical solutions", whose codimension-1 (center-)stable manifold acts as the critical surface separating the basins of attraction of initiation and decay. This understanding was used in sophisticated numerical methods of calculating initiation thresholds, e.g. [21].Here we propose a practical method of defining the initiation criteria analytically. The idea is based...

We consider the problem of ignition of propagating waves in one-dimensional bistable or excitable systems by an instantaneous spatially extended stimulus. Earlier we proposed a method [I. Idris and V. N. Biktashev, Phys. Rev. Lett. 101, 244101 (2008)] for analytical description of the threshold conditions based on an approximation of the (center-)stable manifold of a certain critical solution. Here we generalize this method to address a wider class of excitable systems, such as multicomponent reaction-diffusion systems and systems with non-self-adjoint linearized operators, including systems with moving critical fronts and pulses. We also explore an extension of this method from a linear to a quadratic approximation of the (center-)stable manifold, resulting in some cases in a significant increase in accuracy. The applicability of the approach is demonstrated on five test problems ranging from archetypal examples such as the Zeldovich-Frank-Kamenetsky equation to near realistic examples such as the Beeler-Reuter model of cardiac excitation. While the method is analytical in nature, it is recognized that essential ingredients of the theory can be calculated explicitly only in exceptional cases, so we also describe methods suitable for calculating these ingredients numerically.

Recurrent ischemic priapism is a common complication of sickle cell disease (SCD). We assessed the burden, characteristics, and types of priapism, including sexual dysfunction, in a cohort of men with and those without SCD, to test the hypothesis that sexual dysfunction is more prevalent in men with SCD. In Kano, Nigeria, we conducted a comparative cross-sectional survey that included 500 and 250 men 18 to 40 years of age, with and without SCD, respectively. The survey used the Priapism Questionnaire and the International Index of Erectile Function for sexual function assessment. All eligible participants approached for the study gave informed consent and were enrolled. Stuttering and major priapism were defined based on the average duration of priapism experiences that lasted ≤4 and >4 hours, respectively. The prevalence of priapism was significantly higher in men with SCD than in those without it (32.6% vs 2%; P < .001). Stuttering priapism accounted for 73.6% of the priapism episodes in men with SCD. Nearly 50% of the participants with SCD-related priapism had never sought medical attention for this complication. The majority of the men with SCD-related priapism used exercise as a coping mechanism. Priapism affected the self-image of the men with SCD, causing sadness, embarrassment, and fear. The percentage of the men with SCD who had erectile dysfunction was more than twofold higher than that of those without SCD who had erectile dysfunction (P = .01). The men with SCD had a higher prevalence of priapism and sexual dysfunction than the men without SCD.

We consider the problem of initiation of propagating waves in a one-dimensional excitable fiber. In the FitzHugh-Nagumo theory, the key role is played by "critical nucleus" and "critical pulse" solutions whose ͑center-͒ stable manifold is the threshold surface separating initial conditions leading to propagation and those leading to decay. We present evidence that in cardiac excitation models, this role is played by "critical front" solutions.

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